A233317 - OEIS

%I #13 Feb 16 2025 08:33:21

%S 108,1728,8748,12348,27648,67500,139968,197568,259308,442368,707472,

%T 708588,1000188,1080000,1581228,2239488,3084588,3161088,4148928,

%U 5467500,7077888,7717500,9020268,11319552,11337408,14074668,16003008,17280000,21003948,25299648

%N Integer areas A of the integer-sided triangles such that the inradius and the radius of the three excircles are perfect squares.

%C Subset of A185210.

%C A = sqrt(s*(p-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);

%C The inradius is r = A/s;

%C The radii of the three excircles are r1 = 2*A/(-a+b+c); x2 = 2*A*b/(a-b+c); x3 = 2*A*c/(a+b-c).

%C The areas A of the primitive triangles of sides (a,b,c) are 108, 12348, ...

%C The areas of the nonprimitive triangles of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.

%C The following table gives the first values (A, a, b, c, r, r1, r2, r3) where A is the area of the triangle, a, b, c the integer sides, r, r1, r2 and r3 are respectively the length of the inradius and the radius of the three excircles.

%C +--------+-----+------+------+------+------+------+------+

%C |    A   |  a  |   b  |   c  |   r  |  r1  |  r2  |  r3  |

%C +--------+-----+------+------+------+------+------+------+

%C |    108 |  15 |   15 |   24 |  2^2 |  3^2 |  3^2 |  6^2 |

%C |   1728 |  60 |   60 |   96 |  4^2 |  6^2 |  6^2 | 12^2 |

%C |   8748 | 135 |  135 |  216 |  6^2 |  9^2 |  9^2 | 18^2 |

%C |  12348 |  91 |  280 |  315 |  6^2 |  7^2 | 14^2 | 21^2 |

%C |  27648 | 240 |  240 |  384 |  8^2 | 12^2 | 12^2 | 24^2 |

%C |  67500 | 375 |  375 |  600 | 10^2 | 15^2 | 15^2 | 30^2 |

%C | 139968 | 540 |  540 |  864 | 12^2 | 18^2 | 18^2 | 36^2 |

%C | 197568 | 364 | 1120 | 1260 | 12^2 | 14^2 | 28^2 | 42^2 |

%C | 259308 | 735 |  735 | 1176 | 14^2 | 21^2 | 21^2 | 42^2 |

%C +--------+-----+------+------+------+------+------+------+

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Excircles.html">Excircles</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Exradius.html">Exradius</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Inradius.html">Inradius</a>.

%t nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[area2>0&& IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[Sqrt[area2]/s]] && IntegerQ[Sqrt[2*Sqrt[area2]/(-a+b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a-b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a+b-c)]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

%Y Cf. A185210.

%K nonn,changed

%O 1,1

%A _Michel Lagneau_, Dec 07 2013